would both need to be close to some
number and so both L1 and L2 would need to be close to some number. However, this is
impossible because they are different.

Suppose f is an increasing function defined on

[a,b]

. Show f must be continuous at all
but a countable set of points. Hint:Explain why every discontinuity of f is a jump
discontinuity and

f (x − ) ≡ lim f (y) ≤ f (x) ≤ f (x+ ) ≡ lim f (y)
y→x − y→x+

with f

(x+)

> f

(x− )

. Now each of these intervals

(f (x− ),f (x+ ))

at a point, x where
a discontinuity happens has positive length and they are disjoint. Furthermore, they
have to all fit in

[f (a),f (b)]

. How many of them can there be which have length at
least 1∕n?

Let f

(x,y)

=

2 2
xx2−+yy2

. Find limx→0

(limy→0 f (x,y))

and limy→0

(limx→0 f (x,y))

. If you
did it right you got −1 for one answer and 1 for the other. What does this tell you
about interchanging limits?

The whole presentation of limits above is too specialized. Let D be the domain of a
function f. A point x not necessarily in D, is said to be a limit point of D if B

(x,r)

contains a point of D not equal to x for every r > 0. Now define the concept of limit in
the same way as above and show that the limit is well defined if it exists. That is, if
x is a limit point of D and limy→xf

(x)

= L1 and limy→xf

(x)

= L2, then
L1 = L2. Is it possible to take a limit of a function at a point not a limit
point of D? What would happen to the above property of the limit being well
defined? Is it reasonable to define continuity at isolated points, those points
which are not limit points, in terms of a limit as is often done in calculus
books?

If f is an increasing function which is bounded above by a constant, M, show that
limx→∞f